The quantum teleportation protocol allows a sender to transfer a quantum state to a distant receiver with the use of classical communications by exploiting an entangled pair of states shared between the sender and receiver [1]}, [2]}.
Inspired by the seminal paper [3]}, a distinct type of quantum teleportation protocol, Hayden-Preskill protocol, has been proposed [4]}, [5]} (see Refs. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} for other closely related protocols).
While the information thrown into a black hole whose dynamics is modeled by a Haar random unitary evolution cannot be retrieved by a local measurement, Hayden and Preskill pointed out that the local measurement combined with the Hawking radiation emitted before the information is thrown into a black hole enables the retrieval of the input information [3]}.
The Hawking radiation collected earlier may be interpreted as a part of entangled states shared between the black hole and local observer from the viewpoint of quantum teleportation protocol.
Later, an explicit protocol to reconstruct the input information from the local measurement was given in Ref. [4]}.
This Hayden-Preskill protocol can be interpreted as a teleportation of input information to a distant local observer by leveraging the initially shared entanglement.
Furthermore, it turns out that the Hayden-Preskill protocol works in other class of quantum systems as long as the dynamics is scrambling.
Hence, the success of teleportation in the protocol depends not only on whether entanglement is initially shared between the involved parties but also crucially on how scrambling the quantum evolution is.
Recently, the interpretation of the Hayden-Preskill protocol mentioned above has led to a new diagnostic tool of information scrambling [4]}, [16]}.
Namely, the quality of teleportation with the Hayden-Preskill protocol serves as a proxy of information scrambling.
In the absence of noises and errors, the diagnosis of scrambling by the Hayden-Preskill protocol extracts the same information as one by the averaged OTOC (REF ). However, the protocol has an advantage over the direct measurement of the averaged OTOC when certain noises and errors exist [5]}, [16]}.
Relatedly, the verification of information scrambling in the system coupled to external systems has been studied in for example [8]}, [20]}, [21]}, [22]}, [23]}, [24]}. It is further linked to a new type of phase transition, so called the entanglement phase transition [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}.
